To goose my weight-loss regime, which appears to have stalled once again — though I am holding my own against the “Google 15” — I am adding a daily exercise regimen for the first time, inspired by hundredpushups.com: each day I will do one more push-up than the day before, starting with one today and culminating with a hundred push-ups by my birthday.
If I keep exactly to that plan, the total number of push-ups I’ll do is 5,050 — one today, two tomorrow, three on Wednesday, four on Thursday, and so on. The sum of the first N numbers from 1 through N is, in general,
(N+1) × N/2
an elegant intuitive proof of which is as follows. List the first N numbers, let’s say 6 for this example:
1 2 3 4 5 6
The sum of the two “outer” numbers is 7:
1 2 3 4 5 6
Removing those, the sum of the next two “outer” numbers is also 7:
2 3 4 5
Removing those, the sum of the final pair is also 7:
3 4
That sum — N+1 — is repeated N/2 times, giving rise to the formula
(N+1) × N/2
“Wait a minute,” I hear you say. “What about when N is odd? Then there’s one extra innermost number with no partner.” That’s true. In that case, the number of pairs that add up to N+1 isn’t N/2, it’s only (N-1)/2:
1 2 3 4 5 6 7
Here N is 7, and there are 3 pairs that add up to 8 — 1 and 7, 2 and 6, 3 and 5 — and 4 is all alone in middle. So the sum is:
(N+1) × (N-1)/2 + the middle number
But the middle number is always (N+1)/2, so this becomes:
(N+1) × (N-1)/2 + (N+1)/2
which is the same as
(N+1)/2 × (N-1) + (N+1)/2
which can be read as adding one more (N+1)/2 to a collection of N-1 of them, for a total of N (N+1)/2’s:
(N+1)/2 × N
which is the same as
(N+1) × N/2
which is the same as the original formula above whether N is odd or even. QED.
OK, let’s get this regimen started. Rrrrrnnnnngghh — one. Whew.
In the months preceding my Bar Mitzvah I became determined that, when the day arrived, my skin should be absolutely clear and blemish-free, despite then being in the throes of pubescent acne. To achieve this goal I swore off chocolate and deep-fried food cold-turkey. For something like two or three months I upheld this prohibition (except for one time when I thoughtlessly accepted a few proffered potato chips, then agonized over them for the next several days). The result: it worked! On the day of my Bar Mitzvah my skin positively glowed.
